Computation of Real Radical Ideals by Semidefinite Programming and Iterative Methods

Systems of polynomial equations with approximate real coefficients arise frequently as models in applications in science and engineering. In the case of a system with finitely many real solutions (the $0$ dimensional case), an equivalent system generates the so-called real radical ideal of the system. In this case the equivalent real radical system has only real (i.e., no non-real) roots and no multiple roots. Such systems have obvious advantages in applications, including not having to deal with a potentially large number of non-physical complex roots, or with the ill-conditioning associated with roots with multiplicity. There is a corresponding, but more involved, description of the real radical for systems with real manifolds of solutions (the positive dimensional case) with corresponding advantages in applications. The stable and practical computation of real radicals in the approximate case is an important open problem. Theoretical advances and corresponding implemented algorithms are made for this problem. The approach of the thesis, is to use semidefinite programming (SDP) methods from algebraic geometry, and also techniques originating in the geometry of differential equations. The problem of finding the real radical is re-formulated as solving an SDP problem. This approach in the $0$ dimensional case, was pioneered by Curto \& Fialkow with breakthroughs in the $0$ dimensional case by Lasserre and collaborators. In the positive dimensional case, important contributions have been made of Ma, Wang and Zhi. The real radical corresponds to a generic point lying on the intersection of boundary of the convex cone of semidefinite matrices and a linear affine space associated with the polynomial system. As posed, this problem is not stable, since an arbitrarily small perturbation takes the point to an infeasible one outside the cone. A contribution of the thesis, is to show how to apply facial reduction pioneered by Borwein and Wolkowicz, to this problem. It is regularized by mapping the point to one which is strictly on the interior of another convex region, the minimal face of the cone. Then a strictly feasible point on the minimal face can be computed by accurate iterative methods such as the Douglas-Rachford method. Such a point corresponds to a generic point (max rank solution) of the SDP feasible problem. The regularization is done by solving the auxiliary problem which can be done again by iterative methods. This process is proved to be stable under some assumptions in this thesis as the max rank doesn\u27t change under sufficiently small perturbations. This well-posedness is also reflected in our examples, which are executed much more accurately than by methods based on interior point approaches. For a given polynomial system, and an integer $d \u3e 0$, Results of Curto \& Fialkow and Lasserre are generalized to give an algorithm for computing the real radical up to degree $d$. Using this truncated real radical as input to critical point methods, can lead in many cases to validation of the real radical

Semidefinite Facial Reduction for Low-Rank Euclidean Distance Matrix Completion

The main result of this thesis is the development of a theory of semidefinite facial reduction for the Euclidean distance matrix completion problem. Our key result shows a close connection between cliques in the graph of the partial Euclidean distance matrix and faces of the semidefinite cone containing the feasible set of the semidefinite relaxation. We show how using semidefinite facial reduction allows us to dramatically reduce the number of variables and constraints required to represent the semidefinite feasible set. We have used this theory to develop a highly efficient algorithm capable of solving many very large Euclidean distance matrix completion problems exactly, without the need for a semidefinite optimization solver. For problems with a low level of noise, our SNLSDPclique algorithm outperforms existing algorithms in terms of both CPU time and accuracy. Using only a laptop, problems of size up to 40,000 nodes can be solved in under a minute and problems with 100,000 nodes require only a few minutes to solve

Iterative Universal Rigidity

A bar framework determined by a finite graph $G$ and configuration $\bf p$ in $d$ space is universally rigid if it is rigid in any ${\mathbb R}^D \supset {\mathbb R}^d$. We provide a characterization of universally rigidity for any graph $G$ and any configuration ${\bf p}$ in terms of a sequence of affine subsets of the space of configurations. This corresponds to a facial reduction process for closed finite dimensional convex cones.Comment: 41 pages, 12 figure